2009 SAJC P2 Q3 [Modified]
Given that ${{u}_{r}}=\frac{1}{{{r}^{3}}}$,
(i)
Show that ${{u}_{r-1}}-{{u}_{r+1}}=\frac{6{{r}^{2}}+2}{{{\left( {{r}^{2}}-1 \right)}^{3}}}$.
(ii)
Hence find $\sum\limits_{r=2}^{n}{\frac{3{{r}^{2}}+1}{{{\left( {{r}^{2}}-1 \right)}^{3}}}}$ in terms of $n$.
(iii)
Use your answer to part (ii) to find $\sum\limits_{r=3}^{n}{\frac{3{{\left( r-1 \right)}^{2}}+1}{{{\left[ {{\left( r-1 \right)}^{2}}-1 \right]}^{3}}}}$ in terms of $n$.
Hence find $\sum\limits_{r=3}^{\infty }{\frac{3{{\left( r-1 \right)}^{2}}+1}{{{\left[ {{\left( r-1 \right)}^{2}}-1 \right]}^{3}}}}$.
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